关于投影算子的性质及其在优化问题中应用的研究

The metric projectors and their differential properties play an important role in sensitivity analysis and the algorithms design. Based on the closed form of the metric projectors over two closed convex sets, this project intends to research the applications of the two metric projectors. This study aims at: (1) computing the two metric projectors’ differential properties, including: directional derivative, B-subdifferential and Clarke generalized Jacobian; (2) researching the two metric projectors’ applications in sensitivity analysis, that is, researching the equivalence of the following conditions in the related optimization problems: the strong second order sufficient condition, the strong regularity of KKT point, strong regularity of KKT point, the strong stability of local optimal solution, the nonsingularity of Clarke’s generalized Jacobian of KKT function, and so on; (3) researching the two metric projectors’ applications in augmented Lagrangian method, that is, studying the second order differential properties of augmented Lagrangian function, and studying the the connections of the convergency conditions and the following conditions: the constraint nondegeneracy condition, the strong second order sufficient condition, strict complementary condition. The project can make contributions to further study the differential properties and wide applications of metric projectors over general convex sets, and the project may provide identifiable ground for studying the related theories and althorithms of nonsymmetric matrix optimization.

投影算子及其微分性质在优化问题的理论分析和算法构建中起着至关重要的作用，本项目拟立足于两类闭凸集合上投影算子的闭形式，研究投影算子在优化问题中的应用：（1）计算投影算子的方向导数、B-次微分、Clarke广义Jacobian；（2）研究投影算子的微分性质在优化问题的灵敏性分析中的应用，即研究强二阶充分性条件、局部最优解的强稳定性、KKT点的强正则性、约束非退化性条件、KKT函数Clarke广义Jacobian的非奇异性等一系列条件之间的等价关系；（3）研究投影算子的微分性质在增广Lagrangian方法中的应用，通过研究增广Lagrangian函数的二阶微分性质，刻画增广Lagrangian方法的收敛性条件与约束非退化性条件、强二阶充分性条件、严格互补条件等之间的关联。本项目为进一步研究一般闭凸集合上投影算子的微分性质及其在优化问题中的应用提供了重要的研究途径，并为进一步研究非对称矩阵优化

本项目研究了一类闭凸锥上的相关优化问题包括：（1）闭凸锥的变分几何性质；（2）投影算子的方向导数和B-次微分；（3） 投影算子的微分性质在优化问题 的灵敏性分析中的应用：强二阶.充分性条件、局部最优解的强稳定性、KKT 点的强正则性、约束非退化性条件、KKT 函数.Clarke 广义 Jacobian 的非奇异性等一系列条件之间的关系。发表论文4篇。